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Stable ALS Approximation in the TT-Format for Rank-Adaptive Tensor Completion

AUTHORS

Lars Grasedyck, Sebastian Krämer

ABSTRACT

Low rank tensor completion is a highly ill-posed inverse problem, particularly
when the data model is not accurate, and some sort of regularization is required
in order to solve it. In this article we focus on the calibration of the data model.
For alternating optimization, we observe that existing rank adaption methods do
not enable a continuous transition between manifolds of different ranks. We denote
this flaw as instability (under truncation). As a consequence of this flaw, arbitrarily
small changes in the singular values of an iterate can have arbitrarily large influence
on the further reconstruction. We therefore introduce a singular value based reg-
ularization to the standard alternating least squares (ALS), which is motivated by
averaging in micro-steps. We prove its stability and derive a natural semi-implicit
rank adaption strategy. We further prove that the standard ALS micro-steps are
only stable on manifolds of fixed ranks, and only around points that have what
we define as internal tensor restricted isometry property iTRIP. Finally, we provide
numerical examples that show improvements of the reconstruction quality up to
orders of magnitude in the new Stable ALS Approximation (SALSA) compared to
standard ALS.